On Radical Formula in Modules

Abstract

Abstract. We will state some conditions under which if a quotient of a module M satisfies the radical formula of degree k (s.t.r.f of degree k), so does M. Especially, we will introduce some particular modules M such that M ⊕ M s. Introduction.In this paper all rings are commutative and with identity, all modules are unitary, R denotes a ring and M denotes an R-module. Also, by ‫ގ‬ we mean the set of positive integers and ‫ގ‬ * = ‫ގ‬ ∪ {0, ∞}. A proper submodule P of M is called prime when from rm ∈ P for some r ∈ R and m ∈ M, we can conclude either m ∈ P or rM ⊆ P (see for example [1, 3, 4, 6, 9, 10, 11, 13, 15, 16]). Let (P : M) be the set of all r ∈ R such that rM ⊆ P. If P is a prime submodule, then P = (P : M) is a prime ideal of R and we say that P is P-prime. Recall that if R is an integral domain, then M is called torsion-free, if for every 0 = r ∈ R and 0 = m ∈ M, we have rm = 0. One can easily verify that P is a P-prime submodule of M if and only if

Envelope Dimension of Modules and the Simplified Radical Formula

“…If k = 1 in the above definition, we drop "of degree 1" and simply say that R (weakly) satisfies the s.r.f. This concept was studied in [4].…”

“…Prime ideals of rings play an important role in commutative ring theory, hence many have tried to generalize this concept to modules. A proper submodule P of M is called prime, when from rm ∈ P for some r ∈ R and m ∈ M, we can conclude either m ∈ P or rM ⊆ P (see, for example, [2,4,11,12,14]). Let (P : M) be the set of all r ∈ R such that rM ⊆ P. If P is a prime submodule, then P = (P : M) is a prime ideal of R and we say that P is P-prime.…”

“…To find a similar characterization for the radical of a submodule, the notion of envelope of a submodule was introduced in [11]. The envelope of a submodule N of M, E M (N) (or E(N) if no ambiguity) is the set of all x ∈ M for which, there exist r ∈ R, m ∈ M, and k ∈ N such that x = rm and r k m ∈ N. The envelope of a submodule is not necessarily itself a submodule (see [4,Proposition 2.1]), so we usually use the submodule generated by it, denoted by RE(N).…”

“…In [4] we said that R satisfies the simplified radical formula (s.r.f.) if rad M (0) = E M (0) (in particular, E M (0) is a submodule) for every R-module M. There we proved (see [4,Theorem 2.15]) that a Noetherian ring satisfies the s.r.f. if and only if it is a ZPI-ring (a ring every non-zero ideal of which is a product of prime ideals).…”

A Relationship Between 2-Primal Modules and Modules That Satisfy the Radical Formula

“…If every R-module satisfies the radical formula, then R is also said to satisfy the radical formula. In literature, there has been an intensive study of modules that satisfy the radical formula, see [1,2,9,12,13,17,18] among others. Unlike commutative rings for which √ I = β(I) for any ideal I, not all modules over commutative rings satisfy the radical formula.…”